Inferring Cellular Developmental Time

“From Multivariate to Longitudinal Data”

April 11, 2017

• Motivation
  • Single cell RNA-Seq
• Model Dataset
  • EDA
• Methods of estimating developmental time
  • PCA
  • Probablistic PCA; Bayesian PCA
  • Gaussian Process Latent Variable Modeling

• What happens in a cell such that it becomes a brain, toe, or a heart?
  • When do these decisions get made? “Who” makes them?
    
    
• How do the developmental trajectories of disease (leukemia / schizophrenia) differ from healthy individuals?
  
  
• Can we identify important transition points and the genetic signature underlying them?

\( \forall \) gene \( g \), fit OLS Regression with known timepoint \( t \) per cell–

\[ \log_2(g +1) \sim \beta_0 + \beta_1 t \]

\( \forall \) gene \( g \), fit regression with permuted timepoint \( t^* \) per cell

\[ \log_2(g +1) \sim \beta_0 + \beta_1 t^* \]

\( \forall \) gene \( g \), fit regression with permuted factor timepoint \( t^{**} \)

\[ \log_2(g +1) \sim \beta_0 + \beta_1 t^{**} \]

Given a matrix of \( D \) genes (features) by \( n \) samples in a matrix \( Y \), determine a latent vector \( P \) with dimension \( 1 \) x \( n \) that reflects the (smooth) developmental trajectory of the \( n \) cells from the variance in \( D \) genes.

• \( D \) can be thought of has a higher dimension space, and we want to infer \( d \) (\( d \) < \( D \)) latent variables in the gene data.
  • One of the \( d \) latent variables ideally reflects developmental ordering.

\( Y \) is a \( D \) x \( n \) matrix. Compute the covariance matrix–

\[ \Sigma = E(YY^{T}) - \mu \mu^T \] where \( \mu = E(Y) \)

Then compute the spectral decomposition of \( \Sigma \)

\[ \Sigma a_j = \lambda_j a_j \] for \( j \in (1, ..., D) \)

Then \( a_j \) represent the eigenvectors of the data matrix \( Y \).
Note the ordering of \( j \) is meaningful– \[ \lambda_1 \geq \lambda_2 \geq ... \geq \lambda_D \geq 0 \]

PCA by itself isn't satisfactory…



Ideas for improvements?

1) Quantifiying uncertainty

2) Non-linear latent variable inference

• Next several images taken from slides via Guy Wolf (Yale)

• These slides can be found here

\( Y \) is a \( D \) x \( n \) matrix. \( x \) is a \( d \) x \( n \) matrix (\( d < D \)). Want to relate \( x \) and \( Y \) and assume that the relationship is linear–

\[ Y = Wx + \epsilon \]
where \( W \) is a \( D \) x \( d \) matrix.

By convention (after locating the matrix),

\[ x \sim \mathcal{N} (0, I) \]
where \( I \) is the identity matrix of dimension \( d \) x \( d \).

Specify the error, \[ \epsilon \sim \mathcal{N} (0, \Psi) \]
where we assume \( \Psi \) to be a diagonal matrix \( D \) x \( D \).

• Assuming the diagonal structure of \( \Psi \) means that all observertional correlations is due to the latent variables.
• In other words, \( Y_i \) for \( i \in (1, ..., n) \) are conditionally independent given \( x \)

Then \( Y_i \) for \( i \in (1, ..., n) \),

\[ Y_i | x_i \sim \mathcal{N} (0, \Psi) \]
Then we can integrate out the latent variables–

\[ p( Y_i | W, \Psi)= \int p( Y_i | x_i , W, \Psi) p(x_i) dx_i \]

Under this specification, we can write the MVN distribution for our observations–

\[ Y \sim \mathcal{N} (0, C), C = WW^T + \Psi \]

Young-Whittle Factor Analysis (1950s)

\( \psi_i \) element of the diagonal of \( \Psi \); add constraint \( \psi_i = \sigma^2 \)

Assume \( \sigma^2 \) known, MLE yields same \( W \) as PCA

\[ \mathcal{L} = \tfrac{-N}{2}\{ D\log(2\pi) + \log|C| + \textrm{tr}(C^{-1}\Sigma) \} \]

• \( \alpha_i \) represents the inverse of the variance
  • for large \( \alpha_i \), contribution of latent factors is low.
• Solved by EM / similar algorithm

• GPLVM provide both a probablistic and non-linear latent variable inference structure
  
  • All other published algorithms provide point estimates, difficult to ascertain uncertainty
    
    
• Under this framework, Bayesian priors are allowed, which have been useful in published studies
  
  
• Tools and methods useful in many data analyses contexts (including machine learning)
  • Motivated here by single cell analysis